Monetary Equilibria in a Cash-in-Advance

Economy with Incomplete Financial Markets∗

Jinhui H. Bai† ‡ and Ingolf Schwarz§

JEPS Working Paper No. 05-005http://jeps.repec.org/papers/05-005.pdf

September 2005

Abstract

The general equilibrium model with incomplete financial markets(GEI) is extended by adding fiat money, fiscal and monetary policyand a cash-in-advance constraint. The central bank either pegs theinterest rate or money supply while the fiscal authority sets a Ricardianor a non-Ricardian fiscal plan. We prove the existence of equilibria inall four scenarios. In Ricardian economies, the conditions requiredfor existence are not more restrictive than in standard GEI. In non-Ricardian economies, the sufficient conditions for existence are moredemanding. In the Ricardian economy, neither the price level nor theequivalent martingale measure are determinate.Keywords: Money, Incomplete Markets, Fiscal Policy, Indeterminacy.JEL: D52, E40, E50.

∗We are grateful to John Geanakoplos, Martin Hellwig and Herakles Polemarchakis forhelpful discussions and suggestions. We benefited from comments of Rudiger Bachmann,Truman Bewley, Gaetano Bloise, Marek Weretka and the participants in the seminar atthe Max Planck Institute for Research on Collective Goods in Bonn, the workshop formathematical economics at Yale, the 1st Annual CARESS-Cowles Conference on GeneralEquilibrium Theory and its Applications at Yale, the 14th European Workshop on GeneralEquilibrium Theory in Zurich, the 2nd Asian Workshop on General Equilibrium Theory inTokyo and the 2005 SAET Conference in Vigo. The authors are grateful for the hospitalityof the Max Planck Institute for Research on Collective Goods in Bonn and of the CowlesFoundation at Yale, respectively. The usual disclaimer applies.

†Corresponding Author: Email: jinhui.bai@yale.edu, Phone: +1-203-432-3591, Fax:+1-203-432-5779.

‡Yale University, Department of Economics, 28 Hillhouse Avenue, New Haven, CT06520, USA.

§Max Planck Institute for Research on Collective Goods, Kurt-Schumacher-Str. 10,53113 Bonn, Germany, and CDSEM, University of Mannheim, Germany.

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1 Introduction

In this paper we extend the standard general equilibrium model with in-complete financial markets by introducing fiat money and adding a publicauthority. The latter consists of a fiscal and a monetary authority. Thefiscal authority sets a fiscal plan consisting of taxes, nominal transfers anda debt policy. The monetary authority, or the central bank, creates fiatmoney at zero costs and earns seignorage from its monetary policy. The ac-tions of both authorities are linked by a common public budget constraint.The transactions technology is supposed to be a simple cash-in-advance con-straint. If the nominal interest rates are positive, non-interest bearing fiatmoney is dominated as a store of value by an interest-bearing nominal bond.The demand for money comes from its role to facilitate trade by means ofthe cash-in-advance constraint within the states of the economy.

As argued in the Fiscal Theory of the Price Level (See, e.g., Woodford1995), the introduction of a government which has to meet some budgetconstraint might add additional restrictions on the set of equilibria. It iswell understood that this possibility depends on whether the fiscal policyis of the Ricardian or the non-Ricardian type. Following Woodford (2001),a fiscal policy is called Ricardian if the government budget is satisfied forevery price vector. If the budget is valid only for some prices, it is callednon-Ricardian. In the latter case, the government budget constraint addsadditional restrictions on the equilibrium set.1

We study four important combinations of fiscal and monetary policiesby combining nominal interest rate peg and money supply policy of thecentral bank with a Ricardian and a non-Ricardian fiscal policy. For all thesecases, we prove existence of an equilibrium and characterize its determinacyproperties.

If the fiscal authority follows a Ricardian policy, there exist monetarycompetitive equilibria under assumptions which are close to the standardassumptions in GEI with financial assets. As in the standard GEI modelwithout a central bank and a fiscal authority, the equilibrium in this Ricar-dian framework is not determinate. More precisely, there exists a monetaryequilibrium under a Ricardian fiscal rule for every fixed positive price level

1The idea that a non-Ricardian policy might lead to a determinate equilibrium firstappeared in Dubey and Geanakoplos (1992). They formally prove the generic local unique-ness under a particular non-Ricardian fiscal policy.

2

and for every fixed equivalent martingale measure. This result is true forboth interest rate peg and money supply policy. Under interest rate peg,we argue that the indeterminacy of the price level is purely nominal but theindeterminacy of the martingale measure can be expected to be real sincemarkets are incomplete. Under money supply control, we conjecture thatthe indeterminacy of the price level might also be real.

If the fiscal authority follows a non-Ricardian policy, existence of equi-librium requires more restrictive assumptions as compared to the Ricardiancase. Loosely speaking, the existence of equilibrium requires either highenough gains to trade or positive tax returns. The intuition is that if thefiscal authority fixes nominal transfers at some predetermined and positivelevel, it must earn seignorage or tax returns to be able to balance its bud-get. If taxes are zero, then the gains to trade in the economy must be largeenough to induce some positive seigniorage income for the government. Tomake this argument precise, we use the measure for the gains to trade in-troduced by Dubey and Geanakoplos (1992, 2003a).

Importantly, every obvious degree of indeterminacy we found in the Ri-cardian economy is lost if we assume that the government trades risklessbonds only. Dubey and Geanakoplos (2006) provide a formal proof forgeneric local uniqueness of equilibria under such a fiscal policy. This resultillustrates the role of fiscal policy for the determinacy of the equilibrium.

The main contributions of this paper to the recent literature are thefollowing. First, we show existence and characterize indeterminacy in acash-in-advance economy with incomplete financial market systems and aRicardian fiscal policy.2 Our results extend the previous findings on exis-tence and indeterminacy in Dreze and Polemarchakis (2000), Bloise, Drezeand Polemarchakis (2005), Bloise (2006) and Nakajima and Polemarchakis(2005) under complete markets to incomplete markets. Under a particu-lar non-Ricardian policy, Dubey and Geanakoplos (2003(b)) prove existenceof general competitive equilibria under both interest rate peg and moneysupply control with incomplete markets. They use a strategic market gameapproach to derive their results. The second contribution of our paper isto provide an alternative proof of existence in a non-Ricardian model. Ourmethod to prove existence adopts more traditional techniques of general

2After completing the first draft of this paper, we learned that Gourdel and Triki(2005) independently studied a similar economy under interest rate peg. They obtainresults similar to our Theorems 1 and 3. We will comment on this in Sections 3.3 and 4.3.

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equilibrium analysis, and does not rely on a market game analysis.The paper is organized as follows. In Section 2, we describe the monetary

economy including the government and define the general equilibrium. InSection 3, we present our main results for a Ricardian economy, includingboth interest rate peg and money supply policy. In Section 4, we provide aparallel result for a non-Ricardian economy. In Section 5 we conclude thepaper and give the proofs of all results in the Appendix.

2 The model

2.1 The economy

We study an exchange economy which extends over two dates, the presenttime t = 0 and the future t = 1. The present is known with certainty,but at date 1 there are S possible states of nature which we index withs ∈ S = 1, . . . , S. Including the present, there are S + 1 states of naturelying in the set S∗ := 0, 1, ..., S. At every s ∈ S∗ there are L consumptiongoods which are indexed with l = 1, . . . , L and traded at spot prices psl. Wedenote a consumption plan at state s ∈ S∗ with xs = (xs1, . . . , xsL) ∈ RL

+,an overall consumption plan with x = (x0, . . . , xs, . . . , xS) ∈ R(S+1)L

+ , aprice vector at state s ∈ S∗ with ps = (ps1, . . . , psL) ∈ RL

+ and an overallprice vector with p = (p0, . . . , ps, . . . , pS) ∈ R(S+1)L

+ . All commodities areperishable.

In t = 0, there are asset markets for J ≤ S financial contracts indexedwith j = 1, . . . , J . Each asset is a promise to deliver V j

s ∈ R+ units ofmoney in every state s ∈ S and is traded at price qj in period zero. Thefirst asset is assumed to be a nominal riskless government bond. There isno default and each risky financial asset is in zero net supply. Denote theS × J-matrix of returns with V , the S × (J − 1)-matrix of the returns ofthe risky assets with A, the 1 × J-vector of asset prices with qV and the1 × (J − 1) -vector of asset prices excluding the price of the bond with q.The price of the one period nominal bond between t = 0 and t = 1 is 1

1+r0,

where r0 is the nominal interest rate between t = 0 and t = 1.We will assume that, within every node s ∈ S∗, the asset markets open

before the commodity markets and on the commodity markets, a householdreceives the revenues from selling endowments at the end of the respectivenode. Therefore, in every state s ∈ S in t = 1, a nominal riskfree bond can

4

again be traded to allow agents to borrow against their income which theyreceive at the end of this period. However, there is no uncertainty involvedat this stage, i.e. each state s ∈ S has only one successor state and this stateserves for accounting purposes only. The date of these successor states iscalled accounting period. The price of a bond traded in state s ∈ S is 1

1+rs,

where rs is the nominal interest rate between state s ∈ S and the accountingperiod.

In addition, there is fiat money which can also be held as a store of valuebetween t = 0 and t = 1 and between t = 1 and the accounting period. Weimpose the following general assumption on the structure of the financialassets:

Assumption 1 rank (V ) = J ≤ S. There exists a riskfree asset at eachs ∈ S∗.

2.2 The households

The economy is populated by a finite set I := 1, ..., I of households. Att = 0, the asset markets open first. On this market, the household tradesmoney ni

0 ∈ R+, riskfree government bonds bi0 ∈ R and a portfolio of risky

assets θi ∈ RJ−1. In addition, household i receives a (lump-sum) transferδiH0 from the government, where H0 ∈ R+ is the aggregate transfer fromwhich every household i gets a share δi ∈ R++. Therefore, household i facesthe constraint

bi0

1 + r0+ q · θi + ni

0 = δiH0, (1)

where 11+r0

is the price of the nominal bond. In the goods markets, whichopen next, household i is subject to the following cash-in-advance con-straint:3

p0 · (xi0 − ei

0)+ ≤ ni

0. (2)

The money at the end of t = 0, mi0, is

mi0 =

(ni

0 − p0 · (xi0 − ei

0)+)

+ p0 · (xi0 − ei

0)− (3)

= ni0 − p0 · xi

0 + p0 · ei0.

Combining (1) and (3), we get

p0 · xi0 +

bi0

1 + r0+ q · θi + mi

0 = δiH0 + p0 · ei0. (4)

3We use the usual definition of the negative and the positive part of a vector: x+ :=(. . . , maxxi, 0, . . . ) and x− := (. . . , max−xi, 0, . . . ) so that x = x+ − x−.

5

Equation (4) is the familiar flow budget constraint, which says that the totalexpenditure within one period cannot exceed the total wealth.

From (2) and (3), we get an equivalent formulation of the cash-in-advanceconstraint as

mi0 ≥ p0 · (xi

0 − ei0)−. (5)

We will use this formulation for the transactions technology because it turnsout to be more convenient.

Household i ∈ I pays a tax τ is in state s ∈ S∗. The tax is specified as the

market value of a vector of commodities, τ is ∈ RL

+, i.e. its budgetary impactis ps · τ i

s. The payment of these taxes occurs at the end of state s ∈ S∗ inquestion.

Denoting household i′s quantity of a bond traded at the beginning ofstate s ∈ S in t = 1 with bi

s and the transfer to household i in state s ∈ S

with δiHs. The flow budget constraint then reads

ps ·xis +

bis

1 + rs+mi

s = bi0 +As ·θi +mi

0 + δiHs +ps ·eis−p0 · τ i

0, ∀s ∈ S. (6)

The cash-in-advance constraint is

mis ≥ ps · (xi

s − eis)−, ∀s ∈ S. (7)

In the accounting period following t = 1, the only economic activityis the payment of the debt and of the income tax in state s ∈ S, ps · τ i

s.Therefore, the terminal condition is

0 ≤ bis + mi

s − ps · τ is, ∀s ∈ S. (8)

In the optimal choice, this condition will hold as an equality.For each i ∈ I, define ei := (ei

s)s∈S∗ , τ i := (τ is)s∈S∗ , mi := (mi

s)s∈S∗ andbi := (bi

s)s∈S∗ . The budget set of every household i is the set 4

Bi(p, q, r,H) :=(

xi,mi, bi, θi) ∈ R(S+1)L

+ ×RS+1+ ×RS+1×RJ−1

∣∣

(4)− (8) hold

.

Every household i ∈ I gets utility from consuming in every nodes ∈ S∗ according to a function ui : R(S+1)L

+ → R. We make the follow-ing assumptions on the household sector:

4To save the notation, we suppress the parameters in the notation. The budget setshould always be understood as Bi (p, q, r, H) := Bi

p, q, r, H; ei, τ i, δi

.

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Assumption 2 For each consumer i, the utility function ui is continuous,quasi-concave and strictly increasing.

Assumption 3 Every household has some endowments after tax in everystate, i.e., ∀i ∈ I,

(eis − τ i

s

)> 0 for every s ∈ S∗. Household one has strictly

positive endowments after tax at every node, i.e.(e1 − τ1

) À 0. Aggregateendowments are bounded, i.e.

∑i

(ei − τ i

) ¿ +∞.5

2.3 The government

At each state s ∈ S∗, the government taxes the household and distributestransfers. We denote the total commodity tax by τ := (τs)s∈S∗ ∈ R(S+1)L

+ ,where τs :=

∑Ii=1 τ i

s. The total lump-sum transfer is the vector H :=(Hs)s∈S∗ ∈ RS+1

+ . For simplicity, we assume throughout the paper that thetransfer is distributed according to the shares (δi)i∈I ,

∑i δ

i = 1.The government trades riskfree bonds B = (Bs)s∈S∗ and supplies bal-

ances M = (Ms)s∈S∗ . If Bs > 0 then the government sells bonds and hencethe term represents new indebtedness against the private sector. If Bs < 0,it means the loan to the private sector.

Assumption 4 The government only trades riskless bonds.

This assumption can be justified by an appeal to realism. It has con-sequences for the determinacy of equilibria. We will comment on this inSection 4.3.

It follows that the government budget constraint is

B0

1 + r0+ M0 = H0 (9)

in period zero and

Bs

1 + rs+ Ms + p0 · τ0 = B0 + M0 + Hs, ∀s ∈ S (10)

in period one.In general, the fiscal policy consists of a plan for taxes, transfers and bond

market actions.6 However, in this paper we keep the taxes fixed and restrict5A vector x ∈ Rn satisfies x > 0 if and only if xi ≥ 0, ∀i = 1, . . . , n, and if there is a j

such that xj > 0. Accordingly, x À 0 if and only if xi > 0, ∀i = 1, . . . , n and x ≥ 0 if andonly if xi ≥ 0, ∀i = 1, . . . , n.

6We emphasize that taxes are only included to make the model more general. Not asingle argument, neither related to the existence of a monetary equilibrium nor related tothe (in)determinacy, depends on strictly positive taxes.

7

attention to different transfer policies in combination with bond marketactions. Subject to this restriction we will study four different combinationsof fiscal and monetary policy of the government: the central bank might pegthe interest rate or fix the money supply, while the fiscal authority mightdetermine transfers endogenously or fix them exogenously.

We restrict attention to the following structure. In the case of endoge-nous transfers, the fiscal authority redistributes the seigniorage income andthe tax returns at each state of the economy. The government bonds adjustaccordingly to satisfy the constraints of the government. Bloise and Pole-marchakis (2006) call such a policy a balanced transfer rule. We adopt theirterminology and say that the fiscal policy follows a balanced transfer rule ifit satisfies the following definition:

Definition 1 The balanced transfer fiscal policy determines the vector (H, B)by the functions H (p,M, r) and B (p,M, r), where

Hs (p,M, r) :=rs

1 + rsMs +

ps · τs

1 + rs, ∀s ∈ S∗,

Bs (p,M, r) := psτs −Ms, ∀s ∈ S∗.

The balanced transfer rule says that the government distributes its revenuein every state of the world. Hence the government needs to know the valueof its seigniorage and the market value of its tax returns at the time whenthe transfers are distributed, i.e. in the first subperiod within a state sincethe transfers occur in the asset markets.7

This fiscal policy satisfies the inequality

Bs + Ms − ps · τs ≥ 0, ∀s ∈ S, (11)

which is an admissibility condition for every equilibrium. Indeed, at the endof each state s ∈ S, the outstanding money supply Ms must be sufficientto enable the private sector to pay its taxes and debt service obligations (ifBs < 0) to the government (if Bs > 0, to pay the excess of taxes over the

7Since commodity prices are determined on commodity markets which, however, meetwhen the asset markets are already closed, the specification of such a policy involvesan informational problem. Perhaps the most consistent interpretation is, first, that theasset and the commodity markets in fact meet at the same time but at different placesand, second, that the possible money flow is restricted to the direction from the assetto the commodity markets. The first point implies that the government can observe thecommodity prices for determining its policy, hence resolving the informational problem.The second point implies that the households can use their asset market funds in thecommodity markets, as required.

8

government’s debt service). Obviously, the balanced transfer fiscal policynot only satisfies (11), but it even satisfies

Bs + Ms − ps · τs = 0, ∀s ∈ S. (12)

A little reflection shows that without this equation, there would be no equi-librium. In fact, because preferences are monotone, households choose theirplans so that there is no slack in (8). In any equilibrium, one therefore has∑

i∈I

(bis + mi

s − ps · τ is

)= 0 for all s ∈ S. Since market clearing requires∑

i∈I bis = Bs and

∑i∈I mi

s = Ms, one gets (12). A fiscal policy which failsto provide for the possibility for (12) holding as an equation would not becompatible with equilibrium.

A fiscal policy which fixes transfers in every state of the world exoge-nously will be called fixed transfer fiscal policy. Formally,

Definition 2 The fixed transfer fiscal policy determines the vector (H,B)by the functions H (p,M, r) and B (p,M, r), where

Hs (p,M, r) := Hs,∀s ∈ S∗, where H0 > 0, Hs ≥ 0, ∀s ∈ S,

B0 (p,M, r) := (1 + r0)(H0 −M0

),

Bs (p,M, r) := (1 + rs)(B0 + M0 + Hs −Ms − p0 · τ0), ∀s ∈ S.

Importantly, notice that under this policy the equation (12) does not holdfor some price and interest rate vector.

An alternative approach would postulate (12) directly as part of thegovernment’s budget constraint. This would be appropriate if money wasa kind of debt of the government so that, in fact, the government is underthe legal obligation to withdraw the money that is issued from the system.For ”outside” money, which involves no obligation, this reasoning does notapply.8

Following Woodford (2001), a fiscal policy is said to be Ricardian if itsatisfies the government budget for every vector of prices and interest rates.It is said to be non-Ricardian if the budget is violated for some vector ofprices and interest rates. If (12) is in fact a part of the government’s budgetconstraint, then it becomes clear that the balanced transfer rule is Ricardian,where the fixed transfer rule is non-Ricardian. In the following, we use these

8Note that, with outside money, the government also has to meet a budget constraintin the terminal node: the government debt in the form of bonds must not be larger thanthe tax returns, i.e. Bs − ps · τs ≤ 0. However, this condition will never be a bindingrestriction in our model. This is why we do not mention it explicitly in the text.

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expressions even though we do not interpret (12) as part of the government’sbudget constraint but as a necessary condition for every equilibrium.

2.4 Competitive equilibria

The market clearing condition is specified in the usual way as

I∑

i=1

eis =

I∑

i=1

xis, ∀s ∈ S∗, (13)

Ms =I∑

i=1

mis, ∀s ∈ S∗, (14)

Bs =I∑

i=1

bis, ∀s ∈ S∗, (15)

0 =I∑

i=1

θis, ∀s ∈ S, (16)

where the equation (13), (14), (15), and (16) are commodity, money andasset market clearing conditions, respectively. We write the bond and riskyasset separately since we want to emphasize the difference of the marketsupply in two cases.

The primitives of the economy can be summarized by the vector

E :=(ui, ei, τ i, δi)i∈I , V

.

Definition 3 An equilibrium for the economy E is a tuple

(p, q, r) , (xi,mi, bi, θ

i)i∈I ,(M,B, H

)

such that

(1) (xi,mi, bi, θ

i) maximizes ui(xi) subject to (xi,mi, bi, θi) ∈ Bi(p, q, r, H

).

(2) The actions of the monetary-fiscal authority(M,B, H

)satisfy (9), (10)

and (12).

(3) In every state, markets clear, i.e. (13)-(16) hold.

An equilibrium is said to be monetary if psl < +∞, ∀ s ∈ S∗, l ∈ L.

In the proofs of the theorems given in the following sections, we useanother equilibrium concept which is more tractable for our purposes. It iswell known that agent i’s maximization problem has a solution only if there

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is no arbitrage possibility on the financial markets. Using the results fromHarrison and Kreps (1979), this implies the existence of a strictly positiveprobability measure µ = µss∈S such that qV = µ

1+r0V . µ is called the

equivalent martingale measure.Denote an S-dimensional vector (v1, . . . , vS) by v1. In addition, r1

1+r1:=(

rs1+rs

)s∈S

and 11+r1

:=(

11+rs

)s∈S

. Combine (8) and (6) to get, ∀s ∈ S,

ps ·xis +

ps · τ is

1 + rs+

rs

1 + rsmi

s = bi0 +As · θi +mi

0 + δiHs + ps · eis− p0 · τ i

0. (17)

Substitute the no-arbitrage condition qV =(

11+r0

, q2, . . . , qJ

)= µ

1+r0V

= µ1+r0

(1, A1, . . . , AJ−1)′9 into (4) and plug in (17) for each s ∈ S to get

p0 · xi0 +

r0

1 + r0mi

0 +µ

1 + r0·(

p1 ¤xi1 +

r11 + r1

¤mi1

)= δiH0

+p0 ·(

ei0−

τ i0

1 + r0

)+

µ

1 + r0·(

δiH1+p1 ¤(

ei1−

11 + r1

¤τ i1

)), (18)

where m ¤n := (ms · ns)s∈S∗ . The left hand side is the expenditure in termsof its date−0 value, while the right hand side is the discounted nominalwealth. For household 1, define the complete markets budget set

B1 (p, µ, r,H) =

(x1,m1) ∈ R(S+1)L+ × RS+1

+

∣∣ (5), (7), (18) hold.

.

From the no-arbitrage conditions, the budget sets of agents i ≥ 2 can also beexpressed as depending on µ instead of q. Following Cass (1984) and Duffieand Shafer (1985), in Definition 4 we define a concept of effective monetaryequilibrium.

Definition 4 An effective equilibrium for the economy E is a tuple

(p, µ, r) , (xi, mi, bi, θ

i)i∈I ,(M, B, H

)

such that

(1) For i ≥ 2, (xi, mi, bi, θ

i) maximizes ui(xi) subject to (xi,mi, bi, θi)∈ Bi

(p, µ, r,H

). For i = 1, (x1,m1) maximizes u1(x1) such that

(x1, m1) ∈ B1(p, µ, r, H

), and

(b1, θ

1)

=(B −∑I

i=2 bi,−∑I

i=2 θi).

(2) The actions of the monetary-fiscal authority(M, B, H

)satisfy(9), (10)

and (12).9We use the notation 1 := (. . . , 1, . . . ).

11

(3) In every state, commodity and money markets clear, i.e. (13)-(14) hold.

An effective equilibrium is said to be monetary if psl < +∞, ∀ s ∈ S∗, l ∈ L.

From Definitions 3 and 4, we can immediately see two differences. First,in the effective equilibrium, household 1 is only restricted by the intertempo-ral budget constraint and the cash-in-advance constraint. Second, household1 does not choose

(b1, θ

1)

directly. Instead he chooses a bond demand toclear the bond markets and a demand for the risky assets to clear these assetmarkets.

It is immediate that every effective equilibrium corresponds to an equi-librium as defined in Definition 3. Indeed, it is easy to see that the no-arbitrage conditions determine q given µ and r. To show that a tuple(

p, µ, r), (xi, mi, b

i, θ

i)i∈I ,(M,B, H

)as defined in the effective mon-

etary equilibrium corresponds to a monetary equilibrium, we first need tocheck that the household 1 satisfies the budget equations (4) - (8) and sec-ond that his choice is still optimal in the sequential constraint. The firstproperty follows directly from Walras law.10 To see that household one stillmaximizes his utility, just notice that the sequential constraint is a subset ofthe intertemporal one. Hence, the old consumption vector must be optimalsince it is still feasible under the sequential constraint and it was already op-timal in the larger intertemporal constraint. These arguments are standardand not made explicit here.

3 Monetary equilibria with balanced transfers

3.1 Interest rate peg

If the central bank pegs the nominal interest rate, then the vector r :=rss∈S∗ is fixed at a target value r. To sustain r in the market, the centralbank accommodates money demand. We impose the following assumptionon monetary policy:

Assumption 5 Interest rates are nonnegative and bounded above, 0 ≤ rs <+∞, ∀s ∈ S∗, and the government accommodates money demand, i.e. Ms =∑

i mis for each s ∈ S∗.

A monetary equilibrium with interest rate peg and balanced transferscan now be defined as follows:

10We leave it an exercise to the reader to check these equations.

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Definition 5 A monetary equilibrium with interest rate peg and balancedtransfers is a monetary equilibrium according to Definition 3 with exoge-nously fixed r satisfying Assumption 5 and a fiscal policy rule according toDefinition 1.

In the following theorem, we show that for every fixed price level and forevery fixed martingale measure, there exists a monetary equilibrium whichimplements the interest rate target of the central bank.

Theorem 1 Suppose Assumptions 1 - 5 hold. Fix 0 < c < +∞ and µ À 0,then for every 0 ≤ r ¿ +∞ there exists a monetary equilibrium with interestrate peg and balanced transfers

(p, q, r) , (xi, mi, b

i, θ

i)i∈I ,(M, B, H

)such

that c =∑

l p0l +∑

l∈L,s∈S µspsl and q = µ1+r0

A.

3.2 Money supply control

Under money supply control, the central bank fixes the money supply pro-cess M := (Ms)s∈S∗ at a target value M . If this is the case, we impose

Assumption 6 Under money supply policy, 0 < M s < +∞, ∀s ∈ S∗.

Combining the balanced transfer policy with money supply control sug-gests the following definition:

Definition 6 A monetary equilibrium with money supply control and bal-anced transfers is a monetary equilibrium according to Definition 3 withexogenously fixed M satisfying Assumption 6 and a fiscal policy rule ac-cording to Definition 1.

In the next theorem we show that for every fixed price level and forevery fixed martingale measure, there exists a monetary equilibrium whichimplements a money supply target M of the central bank. So the resultparallels the result from the previous theorem under interest rate peg policy.

Theorem 2 Suppose Assumptions 1 - 4 and 6 hold. Fix 0 < c < +∞ andµ À 0, then for every 0 ¿ M ¿ +∞ there exists a monetary equilibriumwith money supply control and balanced transfers

(p, q, r) , (xi, mi, b

i, θ

i)i∈I ,(M,B, H

)such that c =

∑l p0l +

∑l∈L,s∈S µspsl and q = µ

1+r0A.

13

3.3 Interpretation and literature

We provide some intuition for the existence and the indeterminacy resultsin Theorems 1 and 2. To prove existence of an equilibrium we use similarassumptions as in the GEI-model with nominal assets. The balanced trans-fer rule implies that (12) is true both in and out of equilibrium. Hence thiscondition is an identity which does not add additional restrictions as com-pared to the standard GEI. Note that our equilibrium could be a no-tradeequilibrium in which there is no seigniorage income for the central bank. Inthis case, the government just redistributes potential tax returns among thehouseholds according to their shares (δi)i∈I .

The intuition concerning indeterminacy can be given by counting equa-tions and variables. The macro variables to be determined in the effec-tive equilibrium are the L(S + 1) commodity spot prices, the S − 1 dimen-sional equivalent martingale measure and the S +1 interest rates. There areL(S + 1) equilibrium restrictions coming from commodity market clearingand S + 1 money market clearing equations. Under interest rate peg, thelatter S + 1 equations are identities and the (S + 1) interest rates are fixedexogenously, hence both the equations and the variables cannot be counted.Finally, there is a single Walras law at work. To see this, note that in everyeffective equilibrium, (9), (10) and (12) hold. From this, it is not hard tosee that, in every effective equilibrium,

r0

1 + r0M0 +

p0 · τ0

1 + r0+

µ

1 + r0·(

r11 + r1

¤M1 +1

1 + r1¤(p1¤τ1)

)(19)

= H0 +µ

1 + r0·H1.

From this equation and the fact that household one only faces the intertem-poral budget constraint, we easily infer that one Walras Law is at work.In total, there are S more variables than independent equations, so S isthe degree of total indeterminacy under both interest rate peg and moneysupply control.11

11The same intuition can be given for the original economy. Under interest rate peg, themacro variables to be determined are the commodity prices and the asset prices, whichis a L(S + 1) + (J − 1)-dimensional vector. We have L(S + 1) + (J − 1) market clearingequations for the commodity and the asset markets and (S +1) market clearing equationsfor the bond markets. Including the government budget constraint at every node s ∈ S∗,it follows that there are S +1 Walras laws. The balanced transfer rule implies that (12) isalways true. From this, we infer another S degrees of redundancies. Hence, in total thereare S more variables than equations, which suggests an overall indeterminacy of degree S.

14

Under interest rate peg, among the S dimensions of indeterminacy thereis (at least) one degree of homogeneity involved. Indeed, if agents react toa doubling of the commodity prices by doubling their portfolios and moneydemand, the transfers and money supply will also double by the balancedtransfer rule and money supply adjustment. Hence the allocation is unaf-fected. In this case, it seems reasonable to conjecture that the remainingS − 1 degrees of indeterminacy which are captured by the measure are real.An argument which supports this conjecture is given in Nakajima and Pole-marchakis (2001).

However, the degree of homogeneity is lost in the case of money supplycontrol. Scaling the overall price level cannot be compensated by scalingmoney demand accordingly since equilibrium money supply is fixed. Hencethe indeterminacy captured by the price level might be real. Similarly, achanging price level might change the endogenous transfers which couldalso imply real effects. Finally, as in the case of interest rate peg, the in-determinacy captured by the measure might also be real because a similarargument as under interest rate peg could apply under money supply control.We conclude this short discussion about real indeterminacy by conjecturingthat the real indeterminacy of Ricardian equilibria with incomplete markets,financial assets paying in money and cash-in-advance constraints might bedifferent under money supply control from the real indeterminacy with in-complete markets and nominal assets.12 We emphasize that this is only aconjecture which is worth to be considered formally in a separate paper.

The recent literature on Ricardian economies can be summarized as fol-lows. Dreze and Polemarchakis (2000) and Bloise, Dreze and Polemarchakis(2005) prove existence and indeterminacy under interest rate peg with com-plete asset markets under a finite and an infinite horizon, respectively. Bloise(2006) shows similar results under an infinite horizon with money supplypolicy. We extend this recent literature on Ricardian economies by provingexistence and indeterminacy under both interest rate peg and money supplypolicy with incomplete markets and a finite time horizon.

Notice that our results do not rely on the number of assets. Hence, thesame intuition as given above applies for the case of complete markets. This

Under money supply control, there are S + 1 more variables and equations which clearlyleaves the conclusion unaffected.

12The degree of real indeterminacy of the latter economy was studied by Geanakoplosand Mas-Colell (1989) and Balasko and Cass (1989).

15

is why Bloise, Dreze and Polemarchakis (2005) and Dreze and Polemarchakis(2000) get basically the same results in terms of indeterminacy.

Under interest rate peg policy, Gourdel and Triki (2005) independentlystudied a closely related economy. They obtained a result similar to our The-orem 1. Within the interest rate peg policy, there are two major differencesbetween Gourdel and Triki (2005) and our model. First, in our model theasset markets open before the commodity markets, as in Woodford (1994)and Bloise, Dreze and Polemarchakis (2005). In Gourdel and Triki (2005),the bond market opens before the commodity market, but the latter opensbefore the markets for the risky assets. Second, we use different techniquesto prove our results. In our proof we use a trick introduced by Cass (1984),while they use the method similar to Werner (1985). Our method leads usto characterize the indeterminacy in terms of the total price level and theequivalent martingale measure, while they use the price level within eachstate as the indeterminate variables.

4 Monetary equilibria with fixed transfers

In every equilibrium with fixed transfer, (12) must be true. Plug (12) as afunction of Bs and (9) as a function of B0 into equation (10) to get

rs

1 + rsMs + r0M0 +

ps · τs

1 + rs+ p0 · τ0 = (1 + r0) H0 + Hs, ∀s ∈ S. (20)

This equation is a necessary condition for an equilibrium under fixed trans-fers. Notice that it can only be satisfied if either taxes or seigniorage arestrictly positive. Under zero taxes, the gains to trade in the economy musthence be large enough to induce some positive seigniorage income for thegovernment.

In the proof of the next two Theorems we will impose a gains to tradehypothesis which goes back to Dubey and Geanakoplos (1992, 2003(a),2003(b)). Define the function ζs : RL × R+ → RL by

ζs(ζs, γ) :=

ζsl if ζsl < 0ζsl1+γ otherwise.

A feasible allocation (x−s, es) := (x0, x1, . . . , xS)|xs=esis said to be γ-Pareto

optimal in state s ∈ S at es if there does not exist a trade vector ζs ∈ RIL instate s such that

∑i ζ

is = 0 and, ∀i ∈ I, ei

s + ζis ≥ 0 and ui(xi

0, xi1, . . . , e

is +

16

ζs(ζis, γ), . . . , xi

S) ≥ ui(xi0, x

i1, . . . , e

is, . . . , x

iS) with at least one i ∈ I where

the strict inequality holds. If (x−s, es) is γ-Pareto optimal in state s ∈ S ates, then we equivalently say that there are no gains to γ-diminished trade ins ∈ S at (x−s, es). Accordingly, the gains to trade at (x−s, es) are definedby

γs(x−s, es) := minγ | there are no gains to γ-diminished trade in s ∈ S.

4.1 Interest rate peg

In the fixed transfer case, we assume that the interest rates are strictlypositive.

Assumption 7 Interest rates are strictly positive and bounded above, 0 <rs < +∞, ∀s ∈ S∗. The government accommodates money demand, i.e.Ms =

∑i m

is for each s ∈ S∗.

Combining an interest rate peg policy of the central bank with the fixedtransfer fiscal policy suggests the following definition:

Definition 7 A monetary equilibrium with interest rate peg and fixed trans-fers is a monetary equilibrium according to Definition 3 with exogenouslyfixed interest rates according to Assumption 7 and a fiscal policy rule ac-cording to Definition 2.

To rule out an exploding commodity price path, we need to impose eithera strictly positive taxation or a gains to trade hypothesis. The followingassumption says that if the tax in some state s ∈ S is zero, then the gains totrade in this state exceed the interest rate. Intuitively, the friction causedby the transactions technology still allows for Pareto-improvements at theinitial endowment allocation in the state s ∈ S.

Assumption 8 For every s ∈ S, either τs > 0, or γs(x−s, es) > rs for allfeasible (x−s, es).

The following theorem states that every interest rate target of the centralbank can be embedded in an equilibrium with fixed transfers. Note that wedo not claim any indeterminacy result here.

Theorem 3 Suppose that Assumptions 1 - 4, 7 and 8 hold. For every0 ¿ r ¿ +∞ there exists a monetary equilibrium with interest rate peg andfixed transfers

(p, q, r) , (xi,mi, b

i, θ

i)i∈I ,(M,B, H

).

17

4.2 Money supply control

The definition of equilibrium is straightforward:

Definition 8 A monetary Equilibrium with money supply control and fixedtransfers is a monetary equilibrium according to Definition 3 with exoge-nously fixed money supply satisfying Assumption 6 and a fiscal policy ruleaccording to Definition 2.

For the same reason as in the interest rate peg, we also need to imposea Gains-to-Trade hypothesis for money supply policy.

Assumption 9 For every s ∈ S, either τs > 0, or, for every feasible(x−s, es), γs(x−s, es) > H0+Hs

Ms−H0−Hstogether with M0 ≥ H0 and M s >

H0 + Hs.

The last theorem states the parallel result of Theorem 3 for the case ofmoney supply control of the central bank.

Theorem 4 Suppose Assumptions 1 - 4, 6 and 9 hold. For every 0 ¿M ¿ +∞, there exists a monetary equilibrium with money supply controland fixed transfers

(p, q, r) , (xi, mi, b

i, θ

i)i∈I ,(M,B, H

).

4.3 Interpretation and literature

An intuition for Theorems 3 and 4 can again be given by counting equa-tions and unknowns. As argued in Section 3.3 for the Ricardian economy,we have one Walras law in every effective equilibrium. However, (12) repre-sents another S equilibrium restrictions in the fixed transfer case. In total,commodity prices plus the interest rates plus the martingale measure consti-tute L(S +1)+(S +1)+(S−1) variables which have to be determined. Thecommodity and money market clearing conditions plus the S restrictionsfrom (12) minus the single Walras law add up to L(S + 1) + (S + 1) + S − 1equilibrium restrictions. Under interest rate peg, both the money marketclearing equations and the interest rates can not be counted. Hence, underinterest rate peg and under money supply control, the number of unknownsand restrictions coincides. This is the intuition for why we do not find somedegree of indeterminacy here.13

13The same logic as in Footnote 9 can be applied for the intuition in the original economy.The difference to the argument given in Footnote 9 is that the fixed transfer policy doesnot imply another S degrees of redundancies in the terminal nodes. Hence, the numberof equations now coincides with the number of unknowns.

18

This conclusion follows from the assumption that the government onlytrades riskfree bonds and the fact that the transfers are fixed. Intuitively, thefixed transfers always impose some restrictions, only the number of restric-tions depends on the set of assets the government trades. Our assumptionthat it only trades riskfree assets implies that it enters period one with stateindependent debt. To allow for budget balance, taxes and seigniorage mustalso be independent of the state. Since there are S states, this provides theintuition why there are S additional restrictions. Now suppose there is a fullset of Arrow securities and that the government trades every such security.Then there is only one additional restriction compared to the Ricardian case.Indeed, in this case the only restriction is given directly by (19) because ofthe exogenous transfers (see Bloise, Dreze and Polemarchakis (2005)).

The main contributions to the theoretical literature14 in economies withnon-Ricardian fiscal policies and an active monetary policy are Dubey andGeanakoplos (1992, 2003a, 2003b, 2006). Dubey and Geanakoplos (1992,2003a) consider a one period model with a cash-in-advance constraint, insideand outside money. Dubey and Geanakoplos (2003b) extend this modelto a stochastic economy with incomplete asset markets and a mixed assetstructure. In all papers, they show, among several other results, existenceof the equilibrium. They do so by using a strategic market game approach.Dubey and Geanakoplos (2006) formally prove generic local uniqueness inthe stochastic economy with incomplete asset markets and nominal assets.

We study a similar economy as Dubey and Geanakoplos (2003b, 2006),but to prove existence we basically follow the ideas in Bloise, Dreze andPolemarchakis (2005) by introducing a price determination mechanism inthe fixed point mapping for every price object. This allows us to estab-lish a unified framework to prove existence of equilibrium in all four caseswe consider. In addition, by embedding each equilibrium object into thefixed point mapping, we provide a clear intuition for the mechanism whichdetermines the equilibrium.

Gourdel and Triki (2005) provide a result similar to our Theorem 3 underinterest rate peg policy. In addition to the differences mentioned in Section3.3, there is one more major distinction in this case. While Gourdel and Triki(2005) need strictly positive taxes to establish the existence of a monetaryequilibrium, our result also allows for the possibility of zero taxes provided

14As opposed to the quite huge macroeconomic literature on the Fiscal Theory of thePrice Level.

19

that the economy has sufficiently high gains to trade.

5 Concluding remarks

To conclude the paper, we discuss some directions of future research. First,a different timing of transactions can be considered. One possibility is to usethe cash-in-advance constraint as introduced by Svensson (1985), where thecommodity markets open before the asset markets. This could be a suitableframework to study both the transaction and precautionary demand formoney. However, different from our two-period model, the new timing needsan infinite horizon to support money’s value. Second, it would be interestingto introduce a Baumol-Tobin structure in which households voluntarily holdmoney as a store of value even though other interest bearing bonds coexist.Both existence and determinacy in the Baumol-Tobin economy are open anddifficult questions. Doing so probably requires more than two periods toenrich the potential transaction patterns. In particular, an infinite horizonmodel would be of interest. Finally, the model presented here delivers aunified framework for monetary and fiscal policy within a GEI-economy.Therefore, it would be of interest to study the general equilibrium effects ofchanging monetary policy parameters. Under incomplete financial marketthe effect can be expected to be real, an important feature for policy analysis.Such an analysis would contribute to the old but fundamental debate aboutthe neutrality of money.

6 Appendix

In this appendix we give the proof for the theorems in the main text. Theproofs are organized as follows. First, we define an abstract economy. Sec-ond, we show the properties of the household and aggregate demand. Thenwe prove the results under different monetary-fiscal policy combinations.

6.1 An abstract economy

Define the inverse price level as c := 1Pl p0l+

Ps∈S,l µspsl

and the new pricesby πsl := cµspsl for all s ∈ S and π0l := cp0l. By construction π lies in theunit simplex

∆ :=

π ∈ R(S+1)L

+

∣∣∣∑

s,l

πsl = 1

.

20

Multiply (4) with c and use the no-arbitrage equation q = 11+r0

(µ ·A)to get

π0 · xi0 +

11 + r0

(bi0 + µ ·A · θi

)+ mi

0 = δiH0 + π0 · ei0, (21)

where bi0 := c bi

0, θi := c θi, mi0 := cmi

0 and H0 := cH0. The cash-in-advanceconstraint in t = 0 is

mi0 ≥ π0 ·

(xi

0 − ei0

)−. (22)

Multiply (6) by cµ to get

πs ·xis +

bis

1 + rs+mi

s = µs

(bi0 + As · θi + mi

0 − π0 · τ i0

)+πs ·ei

s +δiHs, (23)

where bis := cµsb

is, mi

s := cµsmis and Hs := c µsHs. The cash-in-advance

constraint at state s ∈ S becomes

mis ≥ πs ·

(xi

s − eis

)−, (24)

and the terminal condition is

bis + mi

s − πs · τ is = 0. (25)

We can now redefine household i’s budget set by

Bi(π, µ, r, H

)=

(xi, mi, bi, θi

)∈ R(S+1)L

+ ×RS+1+ ×RS+1×RJ−1

∣∣

(21)− (25) hold

.

By redefining variables, the intertemporal constraint (18) becomes

π0 · xi0 +

π1 · xi1

1 + r0+

r0

1 + r0mi

0 +1

1 + r0

r11 + r1

· mi1

=δi

(H0 +

H1 · 11 + r0

)+π0 ·

(ei0−

τ i0

1 + r0

)+

11 + r0

π1·(

ei1−

11 + r1

¤τ i1

).(26)

The household 1’s budget constraint is

B1(π, r, H

)=

(x1, m1) ∈ R(S+1)L

+ × RS+1+

∣∣ (22) , (24) and (26) hold

.

It is easy to understand that (9), (10) and (12) are equivalent to (20). Withthe obvious definitions, equation (20) becomes in the abstract economy

rs

1 + rsMs + µsr0M0 +

πs · τs

1 + rs+ µsπ0 · τ0 = (1 + r0)µsH0 + Hs, ∀s ∈ S.(27)

21

The arguments we used to derive (26) can also be used to derive an in-tertemporal reformulation of (9), (10) and (12),

r0

1 + r0M0 +

11 + r0

r11 + r1

· M1 +π0 ·τ0

1 + r0+

11 + r0

π1 ·(

11 + r1

¤τ1

)

= H0 +H1 · 11 + r0

. (28)

A monetary equilibrium in this abstract economy is a vector

(π, µ, r, c) ,

(xi, mi, b

i, θ

i)i∈I ,

(M, B, H

)such that markets clear, households optimize,

(27) is true, c > 0 and µ À 0. Such an equilibrium corresponds to a mone-tary effective equilibrium

(p, µ, r) , (xi, mi, b

i, θ

i)i∈I ,(M,B, H

)according

to Definition 4. As argued earlier, the latter vector corresponds to a mone-tary equilibrium according to Definition 3. In the following proofs, we willtherefore concentrate on equilibria in the abstract economy.

6.2 The household and market demand

µ is an element of the S-dimensional unit simplex, which we denote with∆S−1. The extended positive real line is as usual R+ := R+ ∪ +∞. Westart by deriving the properties of the budget sets in the following lemma:

Lemma 1 Under Assumptions 1 and 3, the budget sets satisfy the followingproperties:

(1.1) For i ≥ 2, Bi(π, µ, r, H

)is a non-empty and upper hemi-continuous

correspondence for(π, µ, r, H

)∈ ∆×∆S−1 × RS+1

+ × RS+1+ .

(1.2) For i ≥ 2, Bi(π, µ, r, H

)is compact for

(π, µ, r, H

)∈ interior(∆)×

interior(∆S−1)× RS+1++ × RS+1

+ .

(1.3) For i ≥ 2, Bi(π, µ, r, H

)is lower hemi-continuous for

(π, µ, r, H

)

∈ interior(∆)×∆S−1 × RS+1+ × RS+1

+ .

(1.4) For i ≥ 2, if H0 > 0 then Bi(π, µ, r, H

)is lower hemi-continuous for(

π, µ, r, H)∈ ∆× interior(∆S−1)× RS+1

+ × RS+1+ .

(1.5) For i ≥ 2, as long as r0 < +∞, Bi(π, µ, r, H

)is lower hemi-continuous

if(π, µ, r, H

)∈ interior(∆)× interior(∆S−1)× RS+1

+ × RS+1+ .

22

(1.6) B1(π, r, H

)is non-empty and upper hemi-continuous for

(π, r, H

)

∈ ∆× RS+1+ × RS+1

+ .

(1.7) B1(π, r, H

)is compact for

(π, r, H

)∈ interior(∆)× RS+1

++ × RS+1+ .

(1.8) B1(π, r, H

)is lower hemi-continuous for

(π, r, H

)∈ ∆×RS+1

+ ×RS+1+ .

(1.9) If H0 > 0 then B1(π, r, H

)is lower hemi-continuous for

(π, r, H

)

∈ ∆× RS+1+ × RS+1

+ .

(1.10) As long as r0 < +∞ and H1 > 0, B1(π, r, H

)is lower hemi-continuous

if(π, r, H

)∈ ∆× RS+1

+ × RS+1+ .

Proof:

(1.1) To check non-emptiness, it is sufficient to notice that(xi, mi, bi, θi

)=

(0, π ¤ ei, 0, 0) satisfies the equations (21)−(25). Upper hemi-continuityis straightforward.

(1.2) Closedness is obvious. To show the boundedness of Bi(π, µ, r, H

)

under (π, µ, r) À 0, note that an action (xi, mi, bi, θi) in Bi(π, µ, r, H

)

must satisfy (26), which implies 0 ≤ (xi, mi) ¿ +∞. From (25) weknow that bi

s > −∞ for every s ∈ S. From the standard no-arbitrageargument, we have −∞ ¿

(bi0, θ

i)¿ +∞. From (23) this further

implies that bis < +∞.

(1.3) To see that there is an interior point, take θi = 0 and for every s ∈ S∗

take xis = 0, bi

s = −2 εis and mi

s = πs · eis + εi

s1+rs

with εis > 0. Using

Assumption 3, π À 0 and r ¿ +∞, it is easy to see that (21) - (25)hold with a strict inequality for all εi

s small enough. Note that thisis true even if µs = 0 for some s ∈ S. This sequence shows that theinterior of the budget set is nonempty. It is now easy to see that theinterior is lower hemi-continuous. Since the closure of a lower hemi-continuous set is again lower hemi-continuous, the result follows.

(1.4) We only need to check that there is an interior point. Change thesequence defined in (1.3) by bi

s = 0 for every s ∈ S∗, mi0 = π0 ·ei

0+ δi eH02

and mis = πs · ei

s + µsδi eH0

4 for every s ∈ S to see that this is true.

23

(1.5) Again, use xi = 0 and θi = 0. In period zero, take bi0 = −2 εi

0,mi

0 = π0 · ei0 + εi

01+r0

with εi0 > 0 and in period one take bi

s = 0 and

mis = πs · ei

s + µsemi

0−π0·τ i0

2 to see that the interior is nonempty for εi0

small enough.

(1.6) It holds that(x1, m1

)= (0, π ¤ e1) is an element of B1

(π, r, H

).

Hence, B1(π, r, H

)is non-empty. The second part is straightforward.

(1.7) This property follows immediately.

(1.8) To see that the interior of B1(π, r, H

)is nonempty, take m1

s = πs ·e1s + εs for every s ∈ S∗, x1 = 0 and choose all εs > 0 small enough.

Note that this argument relies on Assumption 3 and r ¿ +∞.

(1.9) Under the assumption H0 > 0, the same sequence as in (1.8) is aninterior point for εi

s small enough for every s ∈ S∗.

(1.10) Since H1 > 0 and r0 < +∞, the same argument as in (1.8) applies. ¥

The demand correspondence for every consumer type i ≥ 2 is defined tobe

(xi, mi, bi, θi)(π, µ, r, H) := (

xi, mi, bi, θi)∈Bi(π, µ, r, H)

∣∣

(xi, mi, bi, θi)∈argmaxui(xi)

Let ϕi(π, µ, r, H) denote the projection of this demand set onto (xi, mi),ϕi

x(π, µ, r, H) the projection of the latter onto xi and ϕiem(π, µ, r, H) theprojection onto mi. Household i = 1 maximizes his utility by choosing(x1, m1) being an element of B1

(π, r, H

). The demand correspondence is

ϕ1(π, r, H

)and the projections are defined as above. We summarize the

properties of individual demand in the following lemma:

Lemma 2 Under Assumptions 1 - 3, household demand satisfies the fol-lowing properties:

(2.1) For i ≥ 2, ϕi(π, µ, r, H

)is non-empty, compact and convex valued

for(π, µ, r, H

)∈ interior(∆)× interior(∆S−1)× RS+1

++ × RS+1+ .

(2.2) For i ≥ 2, ϕi(π, µ, r, H

)is upper hemi-continuous under the condi-

tions given in Lemma (1.3), (1.4) or (1.5).

24

(2.3) ϕ1(π, r, H

)is non-empty, compact and convex valued for

(π, r, H

)

∈ interior(∆)× RS+1++ × RS+1

+ .

(2.4) ϕ1(π, r, H

)is upper hemi-continuous under the conditions given in

Lemma (1.8), (1.9) or (1.10).

(2.5) Under the assumption of Lemma (1.8), (1.9) or (1.10),

inf‖x‖ ∣∣x ∈ ϕ1

x

(π, r, H

)→ +∞

if πsl → 0 for some s ∈ S∗ and l ∈ L.

(2.6) ∀s ∈ S∗, if rs > 0, then mis ≤ πs · ei

s for all (. . . , mis, . . . )

∈ ϕiem (π, µ, r, H

)if i ≥ 2 and for all (. . . , m1

s, . . . ) ∈ ϕ1em (π, r, H

).

(2.7) For every i ≥ 2, under the conditions of Lemma (1.4) it holds that ifr0 → +∞, then mi

0 → 0 for all (mi0, m

i1 . . . , mi

S) ∈ ϕiem (π, µ, r, H

).

Under the conditions of Lemma (1.9), the same property is true fori = 1 for every (m1

0, m11 . . . , m1

S) ∈ ϕ1em (π, r, H

).

(2.8) For every i ≥ 2, under the conditions of Lemma (1.5) it holds that ifthere is a s′ ∈ S with rs′ → +∞, then mi

s′ → 0 for all

(mi0, . . . , m

is′ , . . . , m

iS) ∈ ϕiem (

π, µ, r, H). Under the conditions of

Lemma (1.10), the same property is true for i = 1 for every(m1

0, m11 . . . , m1

S) ∈ ϕ1em (π, r, H

).

Proof: Parts (2.1) - (2.5) follow from standard arguments using theresults from Lemma 1. Since money is dominated as a store of value for astrictly positive interest rate, mi

s = πs · (xis − ei

s)−, ∀s ∈ S∗, ∀i ∈ I. This

implies (2.6). Concerning (2.7), we first argue for i ≥ 2. For r0 → +∞ weargue that the sequence of best responses converges to a (xi, mi, bi, θi) suchthat mi

0 = 0. From Lemma (2.2), the demand set is upper hemi-continuousalong this sequence. We will argue that if mi

0 > 0 in the limit, then thehousehold can increase his utility. Since the cash-in-advance constraint bindsin the case of positive interest rates, mi

0 > 0 implies that he sells somethingof his endowment. If he deviates by selling nothing and consuming what hesold before, his utility increases. The revenue which he loses in period onefrom not selling the endowment in period zero can be taken from buyingcostless bonds. This deviation implies that mi

0 > 0 cannot be the best

25

response in the limit. For household i = 1, the same property is true. Infact, by optimality, rs

1+rsmi

s = rs1+rs

πs · (xis− ei

s)−. Using this in (26) implies

π0 ·(x1

0 − e10

)+ +π1 ·

(x11 − e1

1

)+

1 + r0= δ1

(H0 +

H1 · 11 + r0

)

+π0 ·

((x1

0 − e10

)−−τ10

)

1 + r0+

11 + r0

π1 ·(((

x11 − e1

1

)−−τ11

)¤ 1

1 + r1

).

This equation reveals that household one earns zero from selling his endow-ments in t = 0. Maximization thus implies selling no endowments. Fromthe cash-in-advance it follows that money demand is zero. Part (2.8) followsfrom the same logic as part (2.7). ¥

Define the market demand correspondence of the commodity and moneyas

Z(π, µ, r, H

):= ϕ1

(π, r, H

)+

∑

i≥2

ϕi(π, µ, r, H

),

and the projections of this set onto commodity and money spaces byZx

(π, µ, r, H

)and Zem (

π, µ, r, H), respectively.

Lemma 3 Under Assumptions 1 - 3, Z(π, µ, r, H

)satisfies the following

properties:

(3.1) Z(π, µ, r, H

)is non-empty, compact and convex-valued for

(π, µ, r, H

)

∈ interior(∆)× interior(∆S−1)× RS+1++ × RS+1

+ .

(3.2) Z(π, µ, r, H

)is upper hemi-continuous for

(π, µ, r, H

)∈ interior(∆)×

∆S−1 × RS+1+ × RS+1

+ .

(3.3) If (zx, zem) ∈ Z(π, µ, r, H

)and if (28) holds, then

(1 + r0) π0 ·(

zx0 −∑

i

ei0

)+ π1 ·

(zx1 −

∑

i

ei1

)

+ r0

(zem0

− M0

)+

r11 + r1

·(zem1

− M1

)= 0.

(3.4) If zem ∈ Zem(π, µ, r, H) then zems ≤ maxs,l

∑i e

isl for r À 0 and all s ∈ S∗.

Proof: Lemma (3.1), (3.2) and (3.4) follow directly from individualdemand (Lemma 2). Lemma (3.3) follows from adding up (26) over i ∈ I

and using (28). ¥

26

6.3 Proof of Theorem 1

We fix the martingale measure µ À 0 and the inverse price level c > 0at the outset. The transfers are determined endogenously according to thebalanced transfer rule.

6.3.1 Preliminary definitions

From Assumption 1, we can define a government transfer function H(π, r, M

)

:=(H0, H1, . . . , HS

)(π, r, M

), where

Hs

(π, r, M

):=

rs

1 + rsMs +

πs · τs

1 + rs, ∀s ∈ S∗.

We slightly abuse the notation by denoting both the function and the imagewith H. By construction, Hs ≥ 0, ∀s ∈ S∗. Obviously, H

(π, r, M

)is a

bounded and continuous function for(π, r, M

)∈ ∆× RS+1

+ × RS+1+ .

6.3.2 Construction of a fixed point mapping

To make the proof compatible with a zero interest rate, we start by definingthe modified interest rate process rn := (rn

s )s∈S∗ by

rns =

rs if rs > 01n if rs = 0.

For n > (S + 1)L define

∆n :=

π ∈ ∆∣∣∣πsl ≥ 1

n

.

It is easy to see that∞∪

n>(S+1)L∆n = interior (∆). Let Kem be a compact

and convex space such that Kem ⊇ Znem (π, H

)for all π ∈ ∆ and H ∈

RS+1+ , where Znem (

π, H)

:= Zem (π, µ, rn, H

). Since rn À 0 for all finite n,

such a compact set exists by Lemma (3.4).15 Define a compact and convexset K eH such that K eH ⊇ Hn

(π, M

)for all π ∈ ∆ and M ∈ Kem, where

Hn(π, M

):= H

(π, rn, M

). Since H is a bounded function and M ∈ Kem,

such a set K eH exists. Further define Znx

(π, H

):= Zx

(π, µ, rn, H

)and a

compact and convex set Knx such that Kn

x ⊇ Znx

(π, H

)for all π ∈ ∆n and

15Even though r depends on n, the set K em does not depend on n by Lemma (3.4).

27

H ∈ K eH . Denote the product set with Kn := Knx × Kem. Note that only

Knx depends on n. Finally, define the mapping

fn : ∆n ×Kem ×K eH ×Kn ⇒ ∆n ×Kem ×K eH ×Kn

by (π, M, H, z

)fn

7→(fn

π , fnfM , fneH , fnz

),

where

fnπ (π, M, H, z) := arg max

π∈∆n

(1 + r0)π0 ·

(zx0−

∑

i

ei0

)+π1 ·

(zx1−

∑

i

ei1

),

fnfM (π, M, H, z) := zem,

fneH(π, M, H, z) := Hn(π, M

),

fnz (π, M, H, z) := Zn

(π, H

).

The first mapping is the price player’s objective function, the second map-ping says that the government accommodates money demand, the thirdmapping is the government transfer function and the last mapping is themarket demand.

From Lemma 3 we infer that fn(π, M, H, z) is a non-empty, compact,convex-valued and upper hemi-continuous correspondence. Kakutani FixedPoint Theorem establishes the existence of a fixed point

(π∗n, M∗n, H∗n, z∗n

).

6.3.3 The limit of the fixed points is an equilibrium

Since(π∗n, M∗n, H∗n, z∗n

)is bounded for each n, we can choose a subse-

quence having a limit(π∗, M∗, H∗, z∗

). π∗ is trivially bounded. By Lemma

(3.4), z∗em is also finite since znem is bounded above by the aggregate endow-ment for all n. By construction, M∗ = z∗em. Since H

(π, M

)is continuous,

H∗ = H(π∗, M∗

). This implies that H∗ is finite. It only remains to show

that z∗x =∑

i ei and z∗x ∈ Zx

(π∗, H∗

).

It follows from M∗ = z∗em and Lemma (3.3) that for all n

(1 + r0) π∗n0 ·(

z∗nx0−

∑

i

ei0

)+ π∗n1 ·

(z∗nx1

−∑

i

ei1

)= 0,

which implies

(1 + r0) π∗0 ·(

z∗x0−

∑

i

ei0

)+ π∗1 ·

(z∗x1

−∑

i

ei1

)= 0

28

in the limit. Consequently, we have z∗x ≤∑

i ei, and z∗x =

∑i e

i if π∗ À 0.However, from Lemma (2.5) we know that household one’s demand goes toinfinity if some π∗sl → 0. Since aggregate excess demand is bounded below,we get that ‖z∗x‖ → +∞ if some π∗sl → 0. Therefore, z∗x ≤

∑i e

i impliesthat π∗ À 0 and z∗x =

∑i e

i. Since Zx

(π, H

)is upper hemi-continuous for

π À 0, we know that z∗x ∈ Zx

(π∗, H∗

).

It is straightforward to see that the vector(π∗, M∗, H∗, z∗

)corresponds

to a monetary equilibrium in the abstract economy under interest rate pegwith balanced transfers.

6.4 Proof of Theorem 2

Similar to the proof of Theorem 1, we fix an arbitrary inverse price levelc > 0 and an arbitrary martingale measure µ À 0.

6.4.1 Preliminary definitions

In the abstract economy, the money supply vector is M = c·(M0,

(µsM s

)s∈S

)

À 0. Define the transfers H(π, r, M

)to individuals as in Section 6.3.1.

Since M is fixed here, we write H (π, r).

6.4.2 Construction of a fixed point mapping

As before, we define ∆n :=π ∈ ∆|πsl ≥ 1

n

for n > (S + 1)L. Define the

set Ωn :=[

1n , n

]S+1 carrying the interest rates r. Let Kem be a compact

and convex space such that Kem ⊇ Zem (π, r, H

)for all π ∈ ∆n, r ∈ Ωn and

H ∈ RS+1+ . Define a set K eH such that K eH ⊇ H (π, r) for all π ∈ ∆n and

r ∈ Ωn. Define the compact and convex set Knx such that Kn

x ⊇ Znx

(π, r, H

)

for all π ∈ ∆n, r ∈ Ωn and H ∈ RS+1+ . The product set is Kn := Kn

x ×Kem.As before, define the mapping

fn : ∆n × Ωn ×K eH ×Kn ⇒ ∆n × Ωn ×K eH ×Kn

by (π, r, H, z

)fn

7→(fn

π , fnr , fneH , fn

z

),

29

where

fnπ

(π, r, H, z

):= arg max

π∈∆n

(1 + r0)π0 ·

(zx0−

∑

i

ei0

)+π1 ·

(zx1−

∑

i

ei1

),

fnr

(π, r, H, z

):= arg max

r∈Ωn

r0

(zem0

− M0

)+

r11 + r1

·(zem1

− M1

),

fneH (π, r, H, z

):= H (π, r) ,

fnz

(π, r, H, z

):= Z

(π, r, H

).

Again, all these mappings satisfy the assumptions required to applyKakutani’s Theorem, implying the existence of a fixed point(π∗n, r∗n, H∗n, z∗n

).

6.4.3 The limit of the fixed points is an equilibrium

Since(π∗n, r∗n, H∗n, z∗n

)is bounded for each n, we can choose a subse-

quence having a limit(π∗, r∗, H∗, z∗

). For every such subsequence,(

π∗, H∗, z∗em)¿ +∞. Since H (π, r) is continuous, we know that H∗ =

H (π∗, r∗). Hence it remains to show that markets clear and (r∗, z∗x) ¿ +∞.Claim 1: In period zero, the money market clears and r∗0 < +∞. To

see this, we argue in three steps.Step 1 : We prove that z∗em0

≤ M0. Suppose not, i.e. z∗em0> M0. From

the construction of fnr , we must have r∗0 = +∞. Then it follows from the

definition of H(π∗, r∗) that H∗0 is strictly positive. Lemma (2.7) implies that

z∗em0= 0, a contradiction to z∗em0

> M∗0 > 0. Hence z∗em0

≤ M0.Step 2 : We prove that z∗em0

= M0 if r∗0 > 0. Since the construction of

fnr implies r∗0

(z∗em0

− M0

)≥ 0 and z∗em0

≤ M0 implies r∗0(z∗em0

− M0

)≤ 0,

we must have r∗0(z∗em0

− M0

)= 0. Therefore, r∗0 > 0 implies z∗em0

− M0 = 0,which means the money market clearing in period zero (with free disposal)in the limit.

Step 3 : We prove r∗0 < +∞. Suppose that r∗0 = +∞. From the firststep, we know that z∗em0

= 0; from the second step, we know z∗em0= M0.

These two facts imply M0 = 0, a contradiction. ¥Claim 2: For every s ∈ S, the money market in state s clears and

r∗s < +∞. To see this, we argue in several steps.Step 1 : We show that z∗x0

≤ ∑i e

i0 and z∗x1

≤ ∑i e

i1. Indeed, using the

definition of H(π, r) and Lemma (3.3), it follows (1+r∗0) π∗0 ·(z∗x0

−∑i e

i0

)+

30

π∗1 ·(z∗x1

−∑i e

i1

)+ r∗0

(z∗em0

− M0

)+ r∗1

1+r∗1·(z∗em1

− M1

)= 0. In Claim 1 we

established r∗0(z∗em0

− M0

)= 0. In addition, the definition of fn

r implies thatr∗1

1+r∗1·(z∗em1

− M1

)≥ 0. From this it follows that (1+r∗0) π∗0 ·

(z∗x0

−∑i e

i0

)+

π∗1 ·(z∗x1

−∑i e

i1

) ≤ 0. From the definition of fnπ we get z∗x0

≤ ∑i e

i0 and

z∗x1≤ ∑

i ei1.

Step 2 : The conditions of Lemma (2.5) apply. To see this, we argue thateither the conditions of Lemma (1.8) or the conditions of Lemma (1.10) aresatisfied. In fact, If r∗1 ¿ +∞, then the conditions of Lemma (1.8) applytrivially. Alternatively, if there is some s ∈ S with r∗s = +∞, then it followsfrom the definition of H(π, r) that H∗

1 > 0. Then the conditions of Lemma(1.10) hold.

Step 3 : We show that π∗ À 0. In fact, we saw in the previous step thatLemma (2.5) can be applied. So if there is a s ∈ S∗ and a l ∈ L such thatπ∗sl = 0, then Lemma (2.5) implies a contradiction to Step 1.

Step 4 : Since r∗0 < +∞ and π∗ À 0, Lemma (2.8) applies. With this inmind, it is easy to see that the Steps 1-3 of Claim 1 apply. Hence Claim 2follows. ¥

From money market clearing it follows now by the same arguments asin the proof of Theorem 1 that z∗x =

∑i e

i. From π∗ À 0 and the upperhemi-continuity of the demand, z∗ ∈ Z(π∗, r∗, H∗).

Finally,(π∗, r∗, H∗, z∗

)corresponds to a monetary equilibrium in the

abstract economy with money supply control and balanced transfers.

6.5 Proof of Theorem 3

With fixed nominal transfers we just introduce a transfer mapping whichtransforms the original transfers into discounted real transfers. In addition,we now determine c and µ endogenously in the fixed point.

6.5.1 Preliminary definitions

Define an augmented taxation τn ∈ R(S+1)L+ as

τns =

τs if τs > 0(

1n , 0, .., 0

) ∈ RL+ if τs = 0

,

where n ∈ N and n > mins

(1

e1s1

). Without loss of generality, if τs = 0 we can

assign the taxation to household 1, i.e. τ1ns =

(1n , 0, .., 0

)and hence τ in

s = 0

31

for all i 6= 1. It is easy to see that household 1 will have a non-empty budgetset for all n > min

s

(1

e1s1

).

Define a government transfer function H(c, µ) :=(H0, H1, . . . , HS

)(c, µ)

by

H0(c, µ) := c H0 and Hs(c, µ) := c µsHs, ∀s ∈ S.

This function is obviously a bounded and continuous function for finite c.Next, define an inverse price level function cn by

cn (π, zem, µ; r) :=r0

1+r0zem0

+ 11+r0

r11+r1

· zem1+ π0·τn

01+r0

+ 11+r0

π1 ·(

11+r1

¤τn1

)

H0 +∑

s∈Sµs

1+r0Hs

,

and use the shortcut cn (π, zem, µ) := cn (π, zem, µ; r). This is a bounded andcontinuous function of (π, zem, µ) for each n as long as H0 > 0 and zem ¿ +∞.Under the latter condition, define the bounded and continuous martingale-measure function µn (π, zem, c, µ) := (µn

1 , . . . , µnS) (π, c, µ, zem; r) by

µns (π, c, µ, zem; r) :=

rs1+rs

zems + 11+rs

πs · τns + µsc

(maxs′∈S

Hs′ −Hs

)

r11+r1

·zem1+π1 ·

(1

1+r1¤τ1

)+c

∑σ∈S µσ

(maxs′∈S

Hs′−Hσ

) .

For fixed r, we just write µns (π, c, µ, zem). By the construction of τn, as long

as π À 0 we have cn > 0, µns > 0 and

∑Ss=1 µn

s = 1 for all finite n.

6.5.2 Construction of a fixed point mapping

Denote aggregate demand with Zn(π, µ, H

):= Z

(π, r, µ, H, τn

). Lemma

(3.4) allows us to define the compact and convex set Kem such that Kem ⊇Znem (

π, µ, H)

for all π ∈ ∆, µ ∈ ∆S−1 and H ∈ RS+1+ . Notice that Kem

does not depend on n. As argued above, for positive H0 we can define acompact and convex set Kn

c such that Knc ⊇ cn (π, zem, µ) for all π ∈ ∆,

zem ∈ Kem and µ ∈ ∆S−1. µn (π, c, µ, zem) lies in a compact and convex setKn

µ ⊂ interior(∆S−1) for π ∈ ∆n, zem ∈ Kem and c ∈ Knc . Introduce the set

KneH such that KneH ⊇ H(c, µ) for all c ∈ Knc and µ ∈ Kn

µ . This set can bechosen to be compact and convex for every n since c ∈ Kn

c . Further defineKn

x such that Knx ⊇ Zn

x

(π, µ, H

)for all π ∈ ∆n, µ ∈ Kn

µ and H ∈ KneH .Finally, Kn := Kn

x ×Kem. Define the mapping

fn : ∆n ×Knc ×Kn

µ ×KneH ×Kn ⇒ ∆n ×Knc ×Kn

µ ×KneH ×Kn

32

by (π, c, µ, H, z

)fn

7→(fn

π , fnc , fn

µ , fneH , fnz

),

where

fnπ

(π, c, µ, H, z

):= arg max

π∈∆n

(1 + r0)π0 ·

(zx0−

∑

i

ei0

)+π1·

(zx1−

∑

i

ei1

),

fnc

(π, c, µ, H, z

):= cn (π, zem, µ) ,

fnµ

(π, c, µ, H, z

):= µn (π, c, µ, zem) ,

fneH (π, c, µ, H, z

):= H(c, µ),

fnz

(π, c, µ, H, z

):= Zn

(π, µ, H

).

fn(π, c, µ, H, z

)is a non-empty, compact, convex-valued and upper hemi-

continuous correspondence. Kakutani fixed point theorem establishes theexistence of a fixed point

(π∗n, c∗n, µ∗n, H∗n, z∗n

).

Note that the money market is always cleared since the central bankaccommodates money demand. From the construction of cn (π, zem, µ), inthe fixed point the equation (28) holds, i.e.

r0M∗n0 +

r11 + r1

·M∗n1 +π∗n0 ·τn

0 +π∗n1 ·(

11 + r1

¤τn1

)= H∗n

0 (1+r0)+H∗n1 ·1.

(29)From the construction of µn (π, zem, c, µ) it follows

r11 + r1

· M∗n1 + π∗n1 ·

(1

1 + r1¤τn

1

)+ c∗n

∑

σ∈S

µ∗nσ

(maxs′∈S

Hs′ −Hσ

)

=

rs1+rs

M∗ns + 1

1+rsπ∗ns · τn

s + c∗nµ∗ns

(maxs′∈S

Hs′ −Hs

)

µ∗ns

.

Use this equation and (29) to get

rs

1 + rsM∗n

s +1

1 + rsπ∗ns ·τn

s −H∗ns +µ∗ns r0M

∗n0 +µ∗ns π∗n0 ·τn

0 = µ∗ns H∗n0 (1+r0),

which proves that (27) holds.

6.5.3 The limit of the fixed points for n → +∞

Since(π∗n, c∗n, µ∗n, H∗n, z∗n

)is bounded for each n, we can let n → +∞

and choose a subsequence having a limit(π∗, c∗, µ∗, H∗, z∗

). We obviously

33

have(π∗, µ∗, z∗em) ¿ +∞, but z∗x could be infinite. Clearly, τn → τ . By

continuity, H∗ = H(c∗, µ∗), c∗ = c(π∗, z∗em, µ∗

)and µ∗ = µ

(π∗, c∗, µ∗, z∗em)

.From z∗em ¿ +∞ we know that c∗ < +∞. From this we infer H∗ ¿ +∞. Itremains to argue that markets clear, (27) holds, 0 ¿ (π∗, c∗, µ∗), z∗x ¿ +∞and z∗ ∈ Z

(π∗, µ∗, H∗

).

Given the construction of cn(π, M, H), Lemma (3.3) applies for ev-ery n. Together with money market clearing (the central bank accommo-dates money demand) it hence follows (1 + r0) π∗n0 · (z∗nx0

−∑i e

i0

)+ π∗n1 ·(

z∗nx1−∑

i ei1

)= 0 for every n. Hence

(1 + r0) π∗0 ·(

z∗x0−

∑

i

ei0

)+ π∗1 ·

(z∗x1

−∑

i

ei1

)= 0.

Since the interest rates are always finite, Lemma (1.8) allows for the ap-plication of the Lemmas (2.4) and (2.5). Hence, z∗x =

∑i e

i ¿ +∞ andπ∗ À 0.

From what we argued above, it follows easily that (27) is also true inthe limit. z∗ ∈ Z

(π∗, µ∗, H∗

)follows from finite interest rates and π∗ À 0

since Lemma (1.3) and Lemma (2.2) apply. Therefore, we need only to showthat c∗ > 0 and µ∗ À 0.

We first prove that µ∗ À 0. Suppose µ∗s = 0 for s ∈ S. From thedefinition of µ(π, m, c, µ) we get zems = 0 and τs = 0. For every n along thesequence of fixed points, the consumer’s budget in s ∈ S is

π∗ns · xi∗ns +

bi∗ns

1 + rs+ mi∗n

s ≤ π∗ns · eis + (bi∗n

0 + Asθi∗n + mi∗n

0 )µ∗ns + δiH∗s ,

mi∗ns ≥ π∗ns · (xi∗n

s − eis)−,

bi∗ns + mi∗n

s = 0.

Since households optimize, we must have rs1+rs

mi∗ns = rs

1+rsπ∗ns · (xi∗n

s − eis)−.

We can use this to derives the equivalent formulation

π∗ns · (xi∗ns − ei

s)+ =

π∗ns · (xi∗ns − ei

s)−

1 + rs+ (bi∗n

0 + Asθi∗n + mi∗n

0 )µ∗ns + δiH∗ns .

By the cash-in-advance constraint, (xi∗s − ei

s)− = 0 for every i ∈ I. Since

markets clear and nobody sells goods it follows that xi∗s = ei

s for all i ∈ I.From H(c, µ), we know that H∗

s = 0. Hence we get (bi∗0 +Asθ

i∗+mi∗0 )µ∗s = 0

from the budget constraint. For n → ∞, we get from the continuity of the

34

budget set and from what we said previously that

π∗s · (xi∗s − ei

s)+ =

π∗s · (xi∗s − ei

s)−

1 + rs.

Define, for every i ∈ I, the utility function vi : RL+ → R by vi(ζi

s) :=ui(xi∗

0 , xi∗1 , . . . , ei

s + ζis, . . . , x

i∗S ). From what we said before, it follows 0 =

arg maxvi(ζi

s)∣∣∣ π∗s · ζi+

s ≤ π∗s ·ζi−s

1+rs

, ∀i ∈ I. Define the function ζs(ζi

s, rs) by

ζs(ζis, rs) :=

ζisl if ζi

sl < 0ζisl

1+rsotherwise

and a utility function virs

(ζis) := vi(ζs(ζi

s, rs)). As argued in Dubey andGeanakoplos (1992, pp. 418-419) we then get the equivalence that 0 =arg max

vi(ζi

s)∣∣∣ π∗s · ζi+

s ≤ π∗s ·ζi−s

1+rs

if and only if 0 = arg max

virs

(ζis)

∣∣∣π∗s ·ζi+

s ≤ π∗s ·ζi−s

. If we consider an economy with I agents having concave

utilities virs

(ζis) and endowments ei

s, then no-trade is a Walrasian equilib-rium for this economy at prices π∗. By Lemma 2 in Dubey and Geanako-plos (2003(a)), at the initial endowment allocation (in state s ∈ S) thereare no gains to rs-diminished trade. Hence, rs ≥ γs(x∗−s, es) from the defi-nition of γs(x∗−s, es) - a contradiction to the Gains-to-Trade Hypothesis inAssumption 8. Therefore, we must have µ∗s > 0 for every s ∈ S.

The definition of c(π, zem, H) and µ(π, zem, c, H) now immediately implyc∗ > 0.

It follows as before that the limit of the fixed point vectors correspondto a monetary equilibrium in the abstract economy with interest rate pegand fixed transfers.

6.6 Proof of Theorem 4

This proof is a combination of the proofs of Theorems 2 and 3.

6.6.1 Preliminary Definitions

The augmented taxation τn ∈ R(S+1)L+ , the government transfer function

H(c, µ) :=(H0, H1, . . . , HS

)(c, µ), the inverse price level function

cn (π, r, µ, zem) and the martingale-measure function µn (π, r, c, µ, zem):= (µn

1 , . . . , µnS) (π, r, c, µ, zem) are defined as in Theorem 3. cn (π, r, µ, zem)

is a bounded and continuous function of (π, r, µ, zem) for each n as long asH0 > 0 and zem ¿ +∞. Under the latter condition, µn (π, r, c, µ, zem) is also

35

bounded and continuous. By the construction of τn, as long as π À 0 wehave cn > 0, µn

s > 0 and∑S

s=1 µns = 1 for all finite n.

6.6.2 Construction of a fixed point mapping

Define ∆n and Ωn as in Theorem 2 and denote aggregate demand withZn

(π, µ, r, H

):= Z

(π, µ, r, H, τn

). Lemma (3.4) allows us to define the

compact and convex set Kem such that Kem ⊇ Znem (π, µ, r, H

)for all π ∈ ∆,

µ ∈ ∆S−1, r ∈ Ωn and H ∈ RS+1+ . As argued above, for positive H0 we

can define a compact and convex set Knc such that Kn

c ⊇ cn (π, r, µ, zem)for all π ∈ ∆n, r ∈ Ωn, µ ∈ ∆S−1 and zem ∈ Kem. µn (π, r, c, µ, zem) lies ina compact and convex set Kn

µ ⊂ interior(∆S−1) for π ∈ ∆n, zem ∈ Kem,

µ ∈ ∆S−1, c ∈ Knc , and zem ∈ Kem. Introduce the compact and convex set

KneH such that KneH ⊇ H(c, µ) for all c ∈ Knc and µ ∈ ∆. Further define

the compact and convex set Knx such that Kn

x ⊇ Znx

(π, µ, r, H

)for all

π ∈ ∆n, µ ∈ Knµ , r ∈ Ωn, and H ∈ KneH . Again, denote the product set by

Kn := Knx ×Kem. Define the mapping

fn : ∆n × Ωn ×Knc ×Kn

µ ×KneH ×Kn ⇒ ∆n × Ωn ×Knc ×Kn

µ ×KneH ×Kn

by (π, r, c, µ, H, z

)fn

7→(fn

π , fnr , fn

c , fnµ , fneH , fn

z

),

where

fnπ

(π, r, c, µ, H, z

):=arg max

π∈∆n

(1 + r0)π0 ·

(zx0−

∑

i

ei0

)+π1·

(zx1−

∑

i

ei1

),

fnr

(π, r, c, µ, H, z

):= arg max

r∈Ωn

r0

(zem0

c−M0

)+

r11 + r1

·(

zem1¤ 1

cµ−M1

),

fnc

(π, r, c, µ, H, z

):= cn (π, r, µ, zem) ,

fnµ

(π, r, c, µ, H, z

):= µn (π, r, c, µ, zem) ,

fneH (π, r, c, µ, H, z

):= H(c, µ),

fnz

(π, r, c, µ, H, z

):= Zn

(π, r, µ, H

),

where 1cµ :=

(1

cµs

)s∈S

in the second line. As before, there exists a fixed

point(π∗n, r∗n, c∗n, µ∗n, H∗n, z∗n

)for every n.

36

6.6.3 The limit of the fixed points is an equilibrium

For n → ∞, we get(π∗n, r∗n, c∗n, µ∗n, H∗n, z∗n

)→

(π∗, r∗, c∗, µ∗, H∗, z∗

)

and τn → τ . We want to show that(π∗, r∗, c∗, µ∗, H∗, z∗

)is an equilibrium

for the abstract economy with taxation τ .By the the definitions of cn (π, r, µ, zem) and µn (π, r, c, µ, zem), we get for

each n

1c∗nµ∗ns

r∗ns

1 + r∗ns

z∗nems+

1c∗nµ∗ns

11 + r∗ns

π∗ns · τns +

1c∗n

(r∗n0 z∗nem0

+ π∗n0 ·τn0

)

= H0(1 + r∗n0 ) + Hs,

or alternatively

1c∗nµ∗ns

r∗ns

1 + r∗ns

z∗nems+

1c∗nµ∗ns

11 + r∗ns

π∗ns · τns +

1c∗n

(r∗n0 z∗nem0

+ π∗n0 ·τn0

)

−r∗n0 H0 = H0 + Hs.

From this,

r∗ns

1 + r∗ns

( 1c∗nµ∗ns

z∗nems− M s

)+ r∗n0

(1

c∗nz∗nem0

−M0

)

+r∗n0

(M0 −H0

)< H0 + Hs − r∗ns

1 + r∗ns

M s,

from which we infer

limn→∞

r∗ns

1 + r∗ns

(1

c∗nµ∗ns

z∗nems−M s

)+ lim

n→∞ r∗n0

(1

c∗nz∗nem0

−M0

)

+ limn→∞ r∗n0

(M0 −H0

) ≤ H0 + Hs − r∗s1 + r∗s

M s.

By the construction of fnr and the fact that M0 > H0, we get in the limit

limn→∞ r∗n0

(1

c∗nz∗nem0

−M0

)≥ 0,

limn→∞

1c∗nµ∗ns

r∗ns

1 + r∗ns

z∗nems≥ r∗s

1 + r∗sM s,

limn→∞ r∗n0

(M0 −H0

) ≥ 0.

Therefore, we haver∗0 < +∞,

since otherwise limn→∞ r∗n0

(M0 −H0

)= +∞, contradicting the inequality.

In addition, since r∗s1+r∗s

M s ≤ H0 + Hs, we know that

r∗s <H0 + Hs

M s −H0 −Hs

< +∞. (30)

37

From r ¿ +∞ and the construction of fnr we can infer that

limn→∞

1c∗n

z∗nem0≤ M0, (31)

limn→∞

1c∗nµ∗ns

z∗nems≤ M s, (32)

which further implies

limn→∞ r∗n0

(1

c∗nz∗nem0

−M0

)= 0, (33)

limn→∞

r∗ns

1 + r∗ns

(1

c∗nµ∗ns

z∗nems−M s

)= 0. (34)

From the definition of cn (π, r, µ, zem) and H(c, µ) we get

r0z∗nem0

+r1

1 + r1· z∗nem1

+π∗n0 ·τn0 +π∗n1 ·

(1

1 + r1¤τn

1

)= H∗n

0 (1+ r0)+ H∗n1 ·1.

Adding up the intertemporal individual budget sets over all households andplugging in this equation gives for every n

(1 + r∗n0 ) π∗n0 ·(

z∗nx0−

∑

i

ei0

)+ π∗n1 ·

(z∗nx1

−∑

i

ei1

)= 0.

The left hand side of this equation is just the commodity price players ob-jective function. In the limit we get

(1 + r∗0) π∗0 ·(

z∗x0−

∑

i

ei0

)+ π∗1 ·

(z∗x1

−∑

i

ei1

)= 0.

Given this, it is easy to see that the commodity markets clear. From Lemma(2.4) we get π∗ À 0. Hence we have z∗ ∈ Z

(π∗, r∗, µ∗, H∗

).

From the construction of cn (π, r, µ, zem), we know that c∗ < +∞. Next,we show that c∗ > 0 and µ∗ À 0.

For µ∗ À 0, the argument is quite similar to the one given in the Theorem3. For every s ∈ S, if τs > 0 , it is obvious that µ∗s > 0. Suppose τs = 0for some s ∈ S and µ∗s = 0. From the fact c∗ < +∞, we know thatlimn→∞ c∗nµ∗ns = 0. From the inequality limn→∞ 1

c∗nµ∗ns

z∗nems≤ M s, we know

that z∗ems= 0 (otherwise limn→∞ 1

c∗nµ∗ns

z∗nems= +∞ > M s). Therefore, the

argument in the proof of Theorem 3 applies, which means γs(x∗−s, es) ≤ r∗s .Hence, by (30), γs(x∗−s, es) < H0+Hs

Ms−H0−Hs, a contradiction to the Gains-to-

Trade hypothesis in Assumption 9. Therefore, we must have µ∗s > 0 for alls ∈ S.

38

The result of c∗ > 0 can be proved in a similar way as in Theorem 3.Given µ∗ À 0 and c∗ > 0 we can now infer from equations (31) - (34)

that the money markets clear.It follows that the limit of the fixed point vectors corresponds to a mon-

etary equilibrium in the abstract economy with money supply control andfixed transfers.

39

References

[1] Balasko, Y., Cass, D., 1989. The structure of financial equilibrium withexogenous yields: The case of incomplete markets. Econometrica, Vol.57, No. 1, 135-162.

[2] Bloise, G., 2006. A note on the existence of monetary equilibrium overan infinite horizon, Economic Theory, 27, 59-77.

[3] Bloise, G., Dreze, J. H., Polemarchakis, H.M., 2005. Monetary equilib-ria over an infinite horizon. Economic Theory 25, 51-74.

[4] Bloise, G., Polemarchakis, H.M., 2006. Theory and practice of monetarypolicy. Economic Theory, 27, 1-23.

[5] Cass, D., 1984. Competitive equilibrium with incomplete financial mar-kets. CARESS working paper 84-09 (University of Pennsylvania).

[6] Dreze, J.H., Polemarchakis, H.M., 2000. Monetary equilibria. in: G.Debreu, W. Neuefeind and W. Trockel, eds., Economic Essays: AFestschrift in Honor of W. Hildenbrand (Springer Verlag) p.p.83-108.

[7] Dubey, P., Geanakoplos, J.D., 1992. The value of money in a finitehorizon Economy: A role for banks, in: P. Dasgupta, D. Gale, D. Hart,and E. Maskin, eds., Economic Analysis of Markets and Games: Essaysin Honor of Frank Hahn. Cambridge, MA: MIT Press, pp. 407–444.

[8] Dubey, P., Geanakoplos, J.D., 2003(a). Inside and outside fiat money,gains to trade, and IS-LM. Economic Theory 21, 347-397.

[9] Dubey, P., Geanakoplos, J.D., 2003(b). Monetary equilibrium withmissing markets. Journal of Mathematical Economics 39, 585-618.

[10] Dubey, P., Geanakoplos, J.D., 2006. Determinacy with nominal assetsand outside money. Economic Theory, 27,.79-106.

[11] Duffie, D., 1987. Stochastic equilibria with incomplete financial mar-kets. Journal of Economic Theory 41, 405-416. Corrigendum 49, 384(1989).

[12] Duffie, D., Shafer, W., 1985. Equilibrium in incomplete market I: Basicmodel of generic existence. Journal of Mathematical Economics 14, 285-300.

40

[13] Geanakoplos, J., Mas-Colell, A., 1989. Real indeterminacy with finana-cial assets. Journal of Economic Theory, Vol. 47, No. 1., 22-38.

[14] Gourdel, P., Triki, L., 2005. Incomplete markets and monetary pol-icy. Working paper B05024, CERMSEM, Universite Paris Pantheon-Sorbonne.

[15] Harrison, M., Kreps, D., 1979, Martingales and arbitrage in multiperiodsecurity markets, Journal of Economic Theory 20, 381 - 408.

[16] Nakajima, T., Polemarchakis, H.M., 2001. Money and prices under un-certainty. Working paper No. 01-32, Brown University.

[17] Nakajima, T., Polemarchakis, H.M., 2005. Money and prices under un-certainty. Review of Economic Studies, Volume 72: Issue 1, 223-246.

[18] Svensson, L., 1985. Money and asset prices in a cash-in-advance econ-omy. Journal of Political Economy 93, 919-944.

[19] Werner, J., 1985. Equilibrium in economies with incomplete financialmarkets. Journal of Economic Theory 36, 110-119.

[20] Woodford, M., 1994. Monetary policy and price level determinacy in acash-in-advance economy. Economic Theory 4, 345-380.

[21] Woodford, M., 1995. Price-level determinacy without control of a mon-etary aggregate, Carnegie-Rochester Conference Series on Public Policy43, 1-46.

[22] Woodford, M., 2001. Fiscal requirements for price stability, Journal ofMoney, Credit and Banking, Vol. 33, No. 3, 669-728.

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